A glimpse into the asymptotics of polynomial identities
28 april 2017
UAntwerpen, Campus Middelheim, G.005 - Middelheimlaan 1 - 2020 Antwerpen (route: UAntwerpen, Campus Middelheim
14 - 15 uur
Seminar by Geoffrey Janssens
About the semniar
Given a set of algebras, a natural problem is to discover which algebras from this set are isomorphic. A classical way to attack such "distinguishing problems" is by means of invariants. In this talk we will associate to any finite dimensional algebra two invariants and be interested in the information they contain. Actually we will do this for the more general class of algebras satisfying a polynomial identity, in short PI algebras. Therefore we will start by an introduction to polynomial identities and the algebraic information they deliver.
More precisely we will explain, for a PI algebra A, the so called codimension sequence, denoted (c_n(A))_n, and some results hereof. Among other, as conjectured by Amitsur and thereafter proved by Berele and Regev, the sequence c_n(A) grows asymptotically as the function f(n) = cn^td^n for some constants c, t and d depending on A. Surprisingly the invariant t is an half-integer and the invariant d even an integer. Moreover, as will be illustrated through examples, these values are computable and tightly connected with the algebraic structure of A. Along the way we will mention results showing that the aforementioned values can be used to distinguish varieties of algebras, in particular certain Morita or more generally derived equivalent algebras. We will also point out the special role played by the representation theory of S_n and the Specht problem (which is a kind of non-commutative version of Noether basis theorem).